\documentclass{kiss} \usepackage{presentation} \usepackage{header} \begin{document} \pagestyle{empty} \hbox{}\vfil \bf \large \hfil Strong-coupling renormalization group\par \smallskip \hfil in a hierarchical Kondo model\par \vfil \hfil Ian Jauslin \rm \normalsize \vfil \small \hfil joint with {\normalsize\bf G.~Benfatto} and {\normalsize\bf G.~Gallavotti}\par \vskip10pt arXiv: \parbox[b]{1cm}{\tt1506.04381\par1507.05678}\hfill{\tt http://ian.jauslin.org/} \eject \pagestyle{plain} \setcounter{page}{1} \title{Kondo model} \begin{itemize} \item [P.~Anderson, 1960], [J.~Kondo, 1964]: $$ H=H_0+V\quad\mathrm{on\ }\mathcal H=\mathcal F_L\otimes\mathbb C^2 $$ \itemptchange{$\scriptstyle\blacktriangleright$} \begin{itemize} \item $H_0$: kinetic term of the {\it electrons} $$ H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(\left(-\frac{\Delta}2-1\right)\,c_\alpha\right)(x)\otimes\mathds1 $$ \item $V$: interaction with the {\it impurity} $$ V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2}c^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c_{\alpha_2}(0)\otimes \tau^j $$ \end{itemize} \itemptreset \end{itemize} \hfil\includegraphics[width=0.8\textwidth]{Figs/kondo_model.pdf}\par \eject \title{Kondo effect: magnetic susceptibility} \begin{itemize} \item Magnetic susceptibility: response to a magnetic field $h$: $$ \chi(h,\beta):=\partial_hm(h,\beta). $$ ($m(h,\beta)$: magnetization). \item Isolated impurity: $$ \chi^{(0)}(0,\beta)=\beta\mathop{\longrightarrow}_{\beta\to\infty}\infty $$ \item Chain of electrons: Pauli paramagnetism: $$ \lim_{\beta\to\infty}\lim_{L\to\infty}\frac1L\chi_e(0,\beta)<\infty. $$ \end{itemize} \eject \title{Kondo effect: magnetic susceptibility} \begin{itemize} \item Turn on the interaction: $\lambda_0\neq0$. Impurity susceptibility $\chi^{(\lambda_0)}(h,\beta)$. \item Ferromagnetic interaction ($\lambda_0>0$): $$ \lim_{\beta\to\infty}\chi^{(\lambda_0)}(0,\beta)=\infty. $$ \item Anti-ferromagnetic interaction ($\lambda_0<0$): $$ \lim_{\beta\to\infty}\chi^{(\lambda_0)}(0,\beta)<\infty. $$ \item {\it Strong-coupling} effect: the qualitative behavior changes as soon as $\lambda_0\neq0$. \end{itemize} \eject \title{Previous results} \begin{itemize} \item [J.~Kondo, 1964]: third order Born approximation. \vskip0pt plus3fil \item [P.~Anderson, 1970], [K.~Wilson, 1975]: renormalization group approach \itemptchange{$\scriptstyle\blacktriangleright$ } \begin{itemize} \item Sequence of effective Hamiltonians at varying energy scales. \item For anti-ferromagnetic interactions, the effective Hamiltonians go to a {\it non-trivial fixed point}. \end{itemize} \itemptreset \item{\tt Remark}: [N.~Andrei, 1980]: the Kondo model (suitably linearized) is exactly solvable via Bethe Ansatz (which breaks down under any perturbation of the model). \end{itemize} \eject \title{Current results} \begin{itemize} \item Hierarchical Kondo model: idealization of the Kondo model that has the same scaling properties. \item It is {\it exactly solvable}: the map relating the effective theories at different scales is {\it explicit} (no perturbative expansions). \item For $\lambda_0<0$, the flow tends to a non-trivial fixed point, and $\chi^{(\lambda_0)}<\infty$ in the $\beta\to\infty$ limit (Kondo effect). \end{itemize} \eject \title{Field theory for the Kondo model} \begin{itemize} \item By introducing an extra dimension ({\it imaginary time}), the partition function $Z:=\mathrm{Tr}(e^{-\beta H})$ can be expressed as the {\it Gaussian} average over a {\it Grassmann} algebra: $$ Z=\mathrm{Tr}\left $$ \vskip-10pt \item Potential: $$ \mathcal V(t)=-\lambda_0\kern-10pt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}}\kern-10pt \psi^+_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^-_{\alpha_2}(t)\tau^j $$ with $\{\psi^\pm_\alpha(t),\psi^\pm_{\alpha'}(t')\}=0$. \item $\left<\cdot\right>$ is defined by its second moment $\left<\psi^-_{\alpha_1}(t_1)\psi^+_{\alpha_2}(t_2)\right>$. \end{itemize} \eject \title{Hierarchical model} \begin{itemize} \item Replace $\psi_\alpha^\pm(t)$ in $\mathcal V(t)$ by $$ \psi_\alpha^\pm(t):=\sum_{m\leqslant0}\psi_\alpha^{[m]\pm}(t) $$ where $\psi_\alpha^{[m]\pm}(t)$ is {\it constant} over the ``time'' intervals $\Delta_{i,\pm}^{[m]}$:\par \vfil \hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxes.pdf}\par \item There are 4 fields in each $\Delta_{i,\pm}^{[m]}$. \item Moments: $$ \left<\psi_\alpha^{[m]-}(\Delta_{i,\mp}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\pm}^{[m]})\right>=\pm 2^m $$ \end{itemize} \eject \title{Full propagator} \begin{itemize} \item Moments: $$ \left<\psi_\alpha^{[m]-}(\Delta_{i,\mp}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\pm}^{[m]})\right>=\pm 2^m $$ \item Full propagator: $$ \left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>= 2^{m_{t,t'}}\mathrm{sign}(t-t') $$ \hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxtree.pdf}\par \vfil \eject \title{Comparison with the Kondo model} \item Hierarchical model: $$ \left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>= 2^{m_{t,t'}}\mathrm{sign}(t-t') $$ \item For the (non-hierarchical) Kondo model: $$ \left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>\approx\sum_m 2^{m}g_\psi^{[0]}(2^m(t-t')) $$ where $g^{[0]}_\psi$ is odd and decays faster than any power. \end{itemize} \eject \title{Hierarchical beta function} \begin{itemize} \item Compute $Z$ by $\mathcal V^{[0]}(t):=\mathcal V(t)$ $$ e^{-\int dt\ \mathcal V^{[m-1]}(t)}:=\left_m $$ \item Effective potential: $$ \int dt\ \mathcal V^{[m]}(t)=\sum_{i=1}^{2^{-m}}\mathcal V^{[m]}_{i,-}+\mathcal V^{[m]}_{i,+} $$ \item Iteration $$ \left_m=\prod_{i=1}^{2^{-m}}\left_m $$ \item By anti-commutation of the fields, $e^{-(\mathcal V_{i,-}^{[m]}+\mathcal V_{i,+}^{[m]})}$ is a polynomial in the fields of order $\leqslant 8$. \end{itemize} \eject \title{Hierarchical beta function} \begin{itemize} \item $\mathcal V^{[m]}$ is parametrized by 2 real numbers ({\it running coupling constants}) $\ell_0^{[m]},\ell_1^{[m]}$: $$\begin{array}{r@{\ }>{\displaystyle}l} \frac{e^{-\scriptstyle\int dt\ \mathcal V^{[m]}(t)}}{C^{[m]}} =&1+\frac{\ell_0^{[m]}}2\int dt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2}} \psi^{[\le m]+}_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^{[\le m]-}_{\alpha_2}(t)\tau^j\\[0.5cm] &+\frac{\ell_1^{[m]}}2\int dt\left(\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2}} \psi^{[\le m]+}_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^{[\le m]-}_{\alpha_2}(t)\right)^2 \end{array}$$ \end{itemize} \eject \title{Hierarchical beta function} \begin{itemize} \item Beta function ({\it exact}) $$\begin{array}{r@{\ }l} C^{[m]}=&\displaystyle1+ \frac32(\ell_0^{[m]})^2+9(\ell_1^{[m]})^2\\[0.3cm] \ell_0^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\ell_0^{[m]} +3 \ell_0^{[m]}\ell_1^{[m]} -(\ell_0^{[m]})^2\Big)\\[0.5cm] \ell_1^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\frac12\ell_1^{[m]}+\frac18(\ell_0^{[m]})^2\Big) \end{array}$$ \end{itemize} \eject \title{Flow} \vfil \hfil\includegraphics[width=0.8\textwidth]{Figs/sd_phase.pdf}\par Fixed points: 0 (stable), $\bm\ell^*$ (marginal in $\ell_0$ and stable in $\ell_1$) \eject \title{Susceptibility} \begin{itemize} \item Add magnetic field $h$ on the impurity. \item New term in the potential: $$ -h \sum_{j\in\{1,2,3\}}\bm\omega_j \tau^j $$ \item 6 running coupling constants. \item The susceptibility can be computed by deriving $C^{[m]}$ with respect to $h$. \end{itemize} \eject \title{Kondo effect} \begin{itemize} \item Fix $h=0$. \item At $0$, the susceptibility diverges as $\beta$. \item At $\bm\ell^*$, the susceptibility remains finite in the $\beta\to\infty$ limit. \end{itemize} \eject \title{Susceptibility} \begin{itemize} \item $\lambda_0=-0.28$ \end{itemize} \hfil\includegraphics[width=200pt]{Figs/sd_susc_0_28.pdf}\par \eject \addtocounter{page}{-1} \title{Susceptibility} \begin{itemize} \item $\lambda_0=-0.02$ \end{itemize} \hfil\includegraphics[width=200pt]{Figs/sd_susc_0_02.pdf}\par \eject \addtocounter{page}{-1} \title{Susceptibility} \begin{itemize} \item $\lambda_0=-0.005$ \end{itemize} \hfil\includegraphics[width=200pt]{Figs/sd_susc_0_005.pdf}\par \eject \title{Open questions} \begin{itemize} \item Magnetic field on the chain as well. This requires defining the hierarchical model to reflect the $x$-dependence of $\psi(x,t)$. \item Rigorous renormalization group analysis for the Kondo model (non-hierarchical). \item The exact solvability of the hierarchical Kondo model is merely a consequence of the fermionic nature of the system. Other fermionic hierarchical models can be studied to investigate other strong-coupling phenomena, e.g. high-$T_c$ superconductivity. \end{itemize} \eject \title{Epilogue: {\tt meankondo}} \begin{itemize} \item The computation in the $h$-dependent case requires computing many Feynman diagrams ($\approx100$). \item Software to perform the computation: {\tt meankondo}. \item {\tt meankondo} can be configured to study any fermionic hierarchical model. \end{itemize} \hfil{\tt http://ian.jauslin.org/software/meankondo/} \end{document}